| L(s) = 1 | + 2-s + 7-s − 8-s − 2·11-s + 6·13-s + 14-s − 16-s + 4·17-s + 12·19-s − 2·22-s + 23-s + 6·26-s + 9·29-s + 2·31-s + 4·34-s + 4·37-s + 12·38-s − 11·41-s + 4·43-s + 46-s − 7·47-s + 7·49-s − 56-s + 9·58-s − 4·59-s + 7·61-s + 2·62-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.377·7-s − 0.353·8-s − 0.603·11-s + 1.66·13-s + 0.267·14-s − 1/4·16-s + 0.970·17-s + 2.75·19-s − 0.426·22-s + 0.208·23-s + 1.17·26-s + 1.67·29-s + 0.359·31-s + 0.685·34-s + 0.657·37-s + 1.94·38-s − 1.71·41-s + 0.609·43-s + 0.147·46-s − 1.02·47-s + 49-s − 0.133·56-s + 1.18·58-s − 0.520·59-s + 0.896·61-s + 0.254·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.418635080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.418635080\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 11 T + 80 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 7 T + 2 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 11 T + 38 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 8 T - 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.661551721970156061428062779576, −9.652709068434193011728793665670, −8.904348762536853763362943191910, −8.534239114370644041235166018900, −8.153927211192121552050246523868, −7.899356743076805658758081521360, −7.28249958751412230256077157485, −7.03641321811563206808671022494, −6.26795048789240915354021246679, −6.12705299990550386674174035839, −5.49168491834556954928837097563, −5.13509841862378702042054862255, −4.94421653299149623475874815360, −4.28798078596763913223665098551, −3.49539953319052853498205150878, −3.46938548816308408760051880023, −2.93027409699647923000437327128, −2.23833209356571198125667153809, −1.11919988156753806813512814014, −1.03038065239971203891007142409,
1.03038065239971203891007142409, 1.11919988156753806813512814014, 2.23833209356571198125667153809, 2.93027409699647923000437327128, 3.46938548816308408760051880023, 3.49539953319052853498205150878, 4.28798078596763913223665098551, 4.94421653299149623475874815360, 5.13509841862378702042054862255, 5.49168491834556954928837097563, 6.12705299990550386674174035839, 6.26795048789240915354021246679, 7.03641321811563206808671022494, 7.28249958751412230256077157485, 7.899356743076805658758081521360, 8.153927211192121552050246523868, 8.534239114370644041235166018900, 8.904348762536853763362943191910, 9.652709068434193011728793665670, 9.661551721970156061428062779576
